Question: Factor the following expression: $4$ $x^2$ $-3$ $x$ $-10$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(4)}{(-10)} &=& -40 \\ {a} + {b} &=& & & {-3} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-40$ and add them together. Remember, since $-40$ is negative, one of the factors must be negative. The factors that add up to ${-3}$ will be your ${a}$ and ${b}$ When ${a}$ is ${5}$ and ${b}$ is ${-8}$ $ \begin{eqnarray} {ab} &=& ({5})({-8}) &=& -40 \\ {a} + {b} &=& {5} + {-8} &=& -3 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {4}x^2 +{5}x {-8}x {-10} $ Group the terms so that there is a common factor in each group: $ ({4}x^2 +{5}x) + ({-8}x {-10}) $ Factor out the common factors: $ x(4x + 5) - 2(4x + 5) $ Notice how $(4x + 5)$ has become a common factor. Factor this out to find the answer. $(4x + 5)(x - 2)$